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A manufacturer drills a hole through the center of a metal sphere of radius
R = 6.The hole has a radius r. Find the volume of the resulting ring in terms of r.
V =

What value of r will produce a ring whose volume is exactly half the volume of the sphere? (Round your answer to two decimal places.)
r=

Sagot :

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Answer:

Step-by-step explanation:

Here's how to find the volume of the ring and the value of r for half the sphere's volume:

1. Volume of the Ring:

Let's denote the volume of the sphere (without the hole) as V_sphere and the volume of the ring as V_ring.

a. Volume of the Sphere:

V_sphere = (4/3)πR^3  (Formula for sphere volume)

Here, R = 6, so V_sphere = (4/3)π (6^3) = 288π

b. Volume of the Removed Cylinder with Hemispheres:

Imagine the removed material as a cylinder with two hemispheres attached at the ends.

i. Volume of Cylinder:

The cylinder's radius is r and its height is equal to the distance between the two flat surfaces created by cutting the sphere (which is 2 times the square root of R^2 - r^2 using the Pythagorean theorem). However, for the ring volume calculation, the height doesn't matter. We'll just denote it as h for now.

Volume of Cylinder = πr^2 * h

ii. Volume of a Hemisphere:

The removed material also includes two hemispheres on both ends of the cylinder. The radius of each hemisphere is also r.

Volume of 1 Hemisphere = (2/3)πr^3

iii. Total Volume Removed:

Total Volume Removed = Volume of Cylinder + 2 * Volume of Hemisphere

= πr^2 * h + 2 * (2/3)πr^3

c. Volume of the Ring:

The volume of the ring is the original sphere's volume minus the volume of the removed material.

V_ring = V_sphere - Total Volume Removed

V_ring = 288π - (πr^2 * h + 2 * (2/3)πr^3)

2. Volume of the Ring as Half the Sphere's Volume:

We want to find the value of r when V_ring = (1/2) * V_sphere

Substitute the expressions for V_ring and V_sphere:

288π - (πr^2 * h + 2 * (2/3)πr^3) = (1/2) * 288π

Simplify and solve for r:

576π - (πr^2 * h + (4/3)πr^3) = 144π

(4/3)πr^3 - πr^2 * h = 432π

Notice that the term with h cancels out because the volume of the ring shouldn't depend on the cylinder's height in this scenario. We are only interested in the radius r.

Therefore:

(4/3)πr^3 - πr^2 * h = 432π  becomes (4/3)πr^3 = 432π

Divide both sides by (4/3)π:

r^3 = 324

Take the cube root of both sides (round to two decimal places):

r ≈ 6.24

Answer:

V_ring = 288π - (πr^2 * h + 2 * (2/3)πr^3) (in terms of r)

r ≈ 6.24 (for the ring volume to be half the sphere's volume)

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